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Numerical Relativity in the World Year of Physics

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					Numerical Relativity in the World Year of Physics


                2005 CAP Congress
                  UBC/TRIUMF
                  Vancouver, BC
                 WE-P4-5, IRC 1
                  June 8, 2005

                  Matthew Choptuik
          CIAR Cosmology & Gravity Program
          Dept of Physics & Astronomy, UBC
                    Vancouver BC
                             Outline


• Trends (since 1995, say)

• Representative (i.e. the best) results
   – 3D GR-hydrodynamics (collapse, NS collisions; Shibata
     and collaborators, PRD [2005])
   – 3D GR-vacuum (BH collisions, i.e. Minkowski vacuum
     doesn’t count, despite the year!!; Pretorius, unpub.)

• Prognosis
                         TRENDS: The Good: Hardware
                     [CFI/ASRA/BCKDF funded HPC infrastructure]
November 1999
                                  vn.physics.ubc.ca
                                  128 x 0.85 GHz PIII, 100 Mbit
                                  Up continuously since 10/98
                                  MTBF of node: 1.9 yrs




                                                        glacier.westgrid.ca
                                                        1600 x 3.06 GHz P4, Gigiabit
March 2005                                              Ranked #54 in Top 500 11/04 (Top in Canada)




 vnp4.physics.ubc.ca
 110 x 2.4 GHz P4/Xeon, Myrinet
 Up continuously since 06/03
 MTBF of node: 1.9 yrs
TRENDS: The Good: Ideas & Algorithms
        Black Hole Excison (Unruh, 1982) &
        Adaptive Mesh Refinement (Berger & Oliger, 1984)
                   TRENDS: The Good

• Community activity

   – 3D vacuum (largely single BH, very slow progress since
     1990 until recent work by Pretorius)

   – 3D matter (in better shape, largely due to lack of
     horizons for much of evolution, as well as weaker
     gravitational fields overall relative to BH)

   – Critical phenomena and other “model problems” continue
     to provide fertile, and arguably the best, training ground
     for GS, PDFs (Liebling, Hirschmann, Gundlach, Lehner,
     Neilsen, Pretorius; Hawke others in the wings)
                    TRENDS: The Good

• Mathematical (incl numerical analytical) maturity

   – Appreciation of importance of hyperbolicity/well-
     posedness … when solving Einstein equations using free
     evolution (too many folk to list)

   – Adoption/adaptation of techniques from numerical
     analysis as a more certain route to stability (LSU group)

   – Successful design and application of constraint dampers
     for free evolution schemes
                       TRENDS: The B…

• Community activity

   – 3D vacuum has been focus of roughly 50% or more of the
     NFS-funded NR effort; to date almost entirely focused on
     SINGLE BH

   – Excrutiatingly slow, and quite predictable, progress since
     1990; no implementation of either of “breakthrough” ideas
     mentioned above

   – Choice of problems studied, who gets funded, funding level,
     has had and continues to have little relation to scientific
     progress; causing resentment among non-N relativists and
     others
                    TRENDS: The Ugly


• Places where we probably don’t want to go, or should
  withdraw from while some of the troops are still standing

   – Solving the binary inspiral problem in corotating
     coordinates

   – Approximately solving Einstein’s equations as an
     INITIAL/BOUNDARY VALUE PROBLEM (IBVP), than as
     a pure INITIAL VALUE PROBLEM (IVP, Cauchy problem)
          TRENDS: The Stark Naked Truth

• Problems we are solving are SIMPLE in specification:
  One page of [tensor] equations, or less;
  In BH-BH case NO PHYSICS OTHER THAN VACUUM GR!!!

   – CAN be “simple” in “implementation”

   – Field dominated, NOT by groups as conventional wisdom
     would have one believe, but by individuals

      • Fluids (Nakamura, Stark, Evans, Shibata, Miller, …)
      • Vacuum (Bruegmann, Pretorius)
      • Critical Phenomena (…)

   – This fact is being ruthlessly exploited by those keeping
     their eyes most firmly fixed on the prize (Pretorius, e.g)
            Representative Results: 3D GR hydro
      (Shibata et al; NS-NS collision; PRD 71:04021 [2005]
            3D core collapse; PRD 71:024014 [2005])
•   3D [x,y,z] (as well as 2D [rho,z], via “Cartoon”) solution of Einstein-
    hydrodynamical equations (fully coupled)

•   Key features of approach

     – BSSN formulation of Einstein equations

     – HRSC treatment of hydro; non trivial EOS (multi parameter,
       “realistic”

     – Single grid, fixed size, but with periodic remap of domain to
       preserve resolution during collapse, a la Evans)
         • 2D: 2,500 x 2,500 x 40,000: 20 h on 4 procs of FACOM VPP5000, BFM1
           (same speed on 8 proc NEC SX6, BFM2)
         • 3D: 440 x 440 x 220 x 15,000: 30 h on 32 processors of BFM1

     – Axisymmetric calcs used in collapse case for hi-res preliminary
       surveys, identifying candidates likely to display “interesting”
       behaviour (e.g. instability) in 3D
         Inspiral and merger of NS-NS binary

• Initial data

   – Irrotational binary stars in quasiequilibrium circular orbits
     (?)

   – Separation slightly larger than “innermost orbit” (where
     Lagrange points appear at the inner edge of the stars)

   – Masses generally chosen in range 1.2 … 1.45 solar

      3 specific cases shown here:
       • 1.30 & 1.30 (equal)
       • 1.25 & 1.35 (unequal)
       • 1.40 & 1.40 (equal)
               Masses: 1.30 and 1.30 solar (equal)




Density contours in x-z plane and lapse visualized.
              Masses: 1.25 and 1.35 solar (unequal)




Density contours in x-z plane and lapse visualized.
               Masses: 1.40 and 1.40 solar (equal)




Density contours in x-z plane and lapse visualized.
     Left: 1.3 & 1.3 Middle: 1.25 & 1.35 Right: 1.4 & 1.4




Density contours in x-z plane and lapse visualized.
    3D core collapse and the development of non-
      axisymmetric instabilities (“bar modes”)
• Initial data
   – Start with axisymmetric code, evolve collapse data (again
     with realistic equation of state), until configuration reaches
     some “strong-gravity” point (lapse < 0.8)
   – Then add l=2 perturbation to excite bar mode instability if
     present

• Key parameter,  , measures how kinetic collapse is, in
  Newtonian theory, ratio of kinetic & grav. potential energies

                               T
                            
                               W
  Bar mode onset in stationary (i.e. non collapsing case) when

                         c  0.27
    3D core collapse and the development of non-
      axisymmetric instabilities (“bar modes”)

• EOS:  ,  : different  above/below nuclear density
        1    2

• Cores shown in the 2.5 – 3.0 solar range

• Initial betas of order 0.001, maximum achieved, order 0.3;
  those configs getting there tend to be oscillating stars above
  nuclear density

• Total gravitational radiation emitted as high as 0.03% of total
  mass, much higher than in axisymmetric collapse
                    Core collapse to NS
      (matter contours in x-z plane; evolution of lapse)




Density contours in x-z plane and lapse visualized.
                    Core collapse to BH
      (matter contours in x-z plane; evolution of lapse)




Density contours in x-z plane and lapse visualized.
                  Comparison of collapse to
                  NS (left) and BH (right)




Density contours in x-z plane and lapse visualized.
               Representative Results: 3D vaccum
                    (Pretorius, unpublished [2005])


•   Key features of approach (in development for about 3 yrs)

    –   “ad hoc”; ignored much “conventional wisdom” (often when CW
        had no empirical basis)

    –   Arguably only fundamentals retained from 30 years of
        cumulative experience in numerical relativity:

        1. Geometrodynamics is a useful concept (Dirac, Wheeler …)
        2. Pay attention to constraints (Dewitt, … )
          Pretorius’s New Code: Key Features

• GENERALIZED harmonic coordinates
• Second-order-in-time formulation and direct discretization
  thereof
• O(h2) finite differences with iterative, point-wise, Newton-
  Gauss-Seidel to solve implicit equations
• Kreiss-Oliger dissipation for damping high frequency solution
  components (stability)
• Spatial compactification
• Implements black hole excision
• Full Berger and Oliger adaptive mesh refinement
• Highly efficient parallel infrastructure (almost perfect
  scaling to hundreds of processors, no reason can’t continue
  to thousands)
• Symbolic manipulation crucial for code generation
           Pretorius’ Generalized Harmonic Code
                   [Class. Quant. Grav. 22, 425, 2005,
              following Garfinkle, PRD, 65:044029, 2002]

•   Adds “source functions” to RHS of harmonic condition


                 
                 x         1
                                g
                                                  
                                              gg  H 

•   Substitute gradient of above into field equations, treat source
    functions as INDEPENDENT functions: retain key attractive feature
    (vis a vis solution as a Cauchy problem) of harmonic coordinates


                        g g ,  ...  0

    Principal part of continuum evolution equations for metric components is
    just a wave operator
        Pretorius’ Generalized Harmonic Code


• Constraints:


                 C   H     x   0


  Can NOT be imposed continuously if source functions are to
  be viewed/treated as independent of the metric functions
Choosing source functions from consideration of
    behaviour of 3+1 kinematical variables

                                         
ds2   2dt 2  hij dx i   i dt dx j   j dt             

H  n  H n  n   ln   K
            i       1
    i
 H  H h              n    i  hij  j ln    ijk h jk
                     


             t   2H  n  ...
             t  i   2  H i  ...
   Choosing source functions from consideration of
       behaviour of 3+1 kinematical variables
• Can thus use source functions to drive 3+1 kinematical vbls
  to desired values

• Example: Pretorius has found that all of the following slicing
  conditions help counteract the “collapse of the lapse” that
  generically accompanies strong field evolution in “pure”
  harmonic coordinates

                        1
               Ht  
                        n
                              1
               t Ht   t  n 
                              
                              1
                  Ht            t Ht
                                 n
                    Constraint Damping
           [Brodbeck et al, J Math Phys, 40, 909 (1999);
                 Gundlach et al, gr-qc/0504114]

• Modify Einstein/harmonic equation via


                         
   g g ,  ...   n C  n C  g n C  0     
  where

                    C   H     x 
                    n    t

• Gundlach et al have shown that for all positive , (to be
  chosen empirically in general), all non-DC contraint-violations
  are damped for linear perturbations about Minkowski
          Merger of eccentric binary systems
                  (Pretorius, work in progress)


• Initial data
   – Generated from prompt collapse of balls of massless
     scalar field, boosted towards each other
   – Spatial metric and time derivative conformally flat
   – Slice harmonic (gives initial lapse and time derivative of
     conformal factor)
   – Constraints solved for conformal factor, shift vector
     components

• Pros and cons to the approach, but point is that it serves to
  generate orbiting black holes
          Merger of eccentric binary systems

• Coordinate conditions

                               1
                    Ht          t Ht
                                 n


                              Hi  0
               ~ 6 / M,        ~ 1 / M,       n5

   – Strictly speaking, not spatially harmonic, which is defined
     in terms of “contravariant components” of source fcns

• Constraint damping coefficient:    ~1/M
               Effect of constraint damping


                                  •   Axisymmetric simulation of
                                      single Schwarzschild hole

                                  •   Left/right calculations
                                      identical except that
                                      constraint damping is used in
                                      right case

                                  •   Note that without constraint
                                      damping, code blows up on a
                                      few dynamical times




Constraint violation visualized
          Representative Results: GR vacuum
 (Merger of eccentric system; Pretorius, unpub. [2005])




Lapse function visualized
            Representative Results: GR vacuum
   (Merger of eccentric system; Pretorius, unpub. [2005])




Scalar field visualized (computational/compactified coords. )
            Representative Results: GR vacuum
   (Merger of eccentric system; Pretorius, unpub. [2005])




Scalar field visualized (uncompactified coords. )
          Representative Results: GR vacuum
 (Merger of eccentric system; Pretorius, unpub. [2005])




“Gravitational radiation” visualized
          Representative Results: GR vacuum
 (Merger of eccentric system; Pretorius, unpub. [2005])




“Gravitational radiation” visualized
          Representative Results: GR vacuum
 (Merger of eccentric system; Pretorius, unpub. [2005])




“Gravitational radiation” visualized
               Computation vital statistics


• Base grid resolution: 48 x 48 x 48
   – 9 levels of 2:1 mesh refinement
       • Effective finest grid 12288 x 12288 x 12288

• Calculation similar to that shown
   – ~ 60,000 time steps on finest level
   – CPU time: about 70,000 CPU hours (8 CPU years)
       • Started on 48 processors of our local P4/Myrinet cluster
       • Continues of 128 nodes of WestGrid P4/gig cluster
   – Memory usage: ~ 20 GB total max
   – Disk usage: ~ 0.5 TB with infrequent output!
                        PROGNOSIS


• The golden age of numerical relativity is nigh, and we can
  expect continued exciting developments in near term
                        PROGNOSIS


• The golden age of numerical relativity is nigh, and we can
  expect continued exciting developments in near term

   – Have scaling issues to deal with, particularly with low-
     order difference approximations in 3 (or more!) spatial
     dimensions; but there are obvious things to be tried
                        PROGNOSIS


• The golden age of numerical relativity is nigh, and we can
  expect continued exciting developments in near term

   – Have scaling issues to deal with, particularly with low-
     order difference approximations in 3 (or more!) spatial
     dimensions; but there are obvious things to be tried

• STILL LOTS TO DO AND LEARN IN AXISYMMETRY AND
  EVEN SPHERICAL SYMMETRY!!
        APS Metropolis Award Winners
(for best dissertation in computational physics)


     1999             LUIS LEHNER

    2000               Michael Falk

     2001               John Pask

    2002               Nadia Lapusta

    2003            FRANS PRETORIUS

    2004               Joerg Rottler

    2005            HARALD PFEIFFER

				
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posted:9/7/2012
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